Asymptotic Analysis - Volume 10, issue 3

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ISSN 0921-7134 (P)
ISSN 1875-8576 (E)

Impact Factor 2018: 0.748

The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.

Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.

Abstract: We determine the Γ-limit as ε→0 of the sequence ∫Ω [εϕ2 (x,∇u)+ε−1 u2 (1−u2 )]dx of functionals defined on H1 (Ω). The free energy density ϕ has linear growth, is convex and positively homogeneous of degree one in the second variable and is upper semi-continuous in the first variable. The Γ-limit can be interpreted as a generalized total variation with discontinuous coefficients on the class of the characteristic functions of sets of finite perimeter. The proof is based on some variational properties of the generalized total variation strictly related to the upper semi-continuity of ϕ and on constructing a suitable…approximation of ϕ, from above.
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Abstract: We build aN-particle stochastic system subject to a multibody interaction. The postcollisional repartition is performed so that the one-particle density function solves a scalar conservation law in a suitable limit of hydrodynamical type. In the kinetic limit, we recover a relaxation model which approximates the scalar. conservation. law. Our proof relies on a direct estimate of the L1 distance between the one-particle density function and its kinetic limit, using an appropriate joint process.

Abstract: We consider a Nernst-Planck-Poisson system modelling ion migration through biological membranes, in the one dimensional case. The model includes both the effects of biochemical reaction between ions and of fixed charges. We state the existence of solutions under either an imposed potential condition or an imposed current condition. We study the asymptotical behaviour of solutions in the limit of electroneutrality. Non-uniform convergence gives rise to jump in the potential, known as Donnan potential. Finally we give correctors which describe the charged boundary layers.